1. Perturbation Theory
Lecture 5

2. Interactions of radiations with matter
In absence of external perturbation
------(1)
In presence of radiations, Hamiltonian will be
(2)

3. In absence of electrostatic source
Also, using Coulomb Gauge condition
Neglecting terms of order of
Eq (2) become
------(3)
----(4)
To find the effect of external perturbation on atom, we
first need to find vector potential
can be calculated classically and quantum
mechanically.
External perturbation

4. Classical treatment of radiations
We write
-----(5)
Polarization vector
E.F. and M.F. are given by
------(6)

5. Energy density for single photon is given by
(7)
Averaging over time and writing energy of single photon
Per unit volume,
Which gives,
(8)
Thus, from (5), we have
(9)

6. Thus, using (9) in (4), Potential is given by
-----(10)
which is like harmonic perturbation.

7. Transition rates for perturbation defined by eq. (10)
are given by
---(11)
(12)
Exercise: Derive above equations.

8. Transition rate within dipole approximation
Expanding , we get
(13)
Reason: is small quantity because wavelength
of radiations (visible or ultraviolet is large) compared
to atomic size
(small quantity)

9. Neglecting higher order terms in (13) i.e. Considering only
first term
(14)
This is called dipole approximation.
We know
Above eq can be generalized to
Which give
(15)

10. Using (14) and (15) in following, we get
(16)
Using (16) in (11), we obtain
transition rates within dipole approximation.

11. Exercise: Discuss selection rules for dipole transitions.
e.g. 2p to 1s transition will be allowed but 2s to 1s will be forbidden.